While the error term is r n x f n 1 ξ n 1 x n 1 e ξ

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, while the error term is R n ( x ) = f ( n +1) ( ξ ) ( n + 1)! x n +1 = e ξ ( n + 1)! x n +1 . We want this to be valid for all x [0 , 0 . 5], so we use x n +1 0 . 5 n +1 , and e ξ e 0 . 5 (since the exponential function is increasing). Hence we have R n ( x ) e 0 . 5 (0 . 5) n +1 ( n + 1)! , and we want this to be smaller than 10 - 6 . The only good way to do this is trial and error. Eventually we find that when n = 7, we have R 7 ( x ) e 0 . 5 (0 . 5) 8 8! < 2 × 10 - 7 . 3
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